# Properties

 Label 4025.2.a.r Level 4025 Weight 2 Character orbit 4025.a Self dual Yes Analytic conductor 32.140 Analytic rank 0 Dimension 5 CM No Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$4025 = 5^{2} \cdot 7 \cdot 23$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 4025.a (trivial)

## Newform invariants

 Self dual: Yes Analytic conductor: $$32.1397868136$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.122821.1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( 1 - \beta_{1} ) q^{2}$$ $$+ ( 1 + \beta_{3} ) q^{3}$$ $$+ ( \beta_{3} + \beta_{4} ) q^{4}$$ $$+ ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{6}$$ $$+ q^{7}$$ $$+ ( -\beta_{2} + \beta_{3} + \beta_{4} ) q^{8}$$ $$+ ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ ( 1 - \beta_{1} ) q^{2}$$ $$+ ( 1 + \beta_{3} ) q^{3}$$ $$+ ( \beta_{3} + \beta_{4} ) q^{4}$$ $$+ ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{6}$$ $$+ q^{7}$$ $$+ ( -\beta_{2} + \beta_{3} + \beta_{4} ) q^{8}$$ $$+ ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{9}$$ $$+ ( -2 + 2 \beta_{2} - \beta_{3} ) q^{11}$$ $$+ ( 2 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{12}$$ $$+ ( \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{13}$$ $$+ ( 1 - \beta_{1} ) q^{14}$$ $$+ ( 1 - 2 \beta_{1} - 2 \beta_{2} - \beta_{4} ) q^{16}$$ $$+ ( 2 - 2 \beta_{1} - 3 \beta_{3} - 2 \beta_{4} ) q^{17}$$ $$+ ( 3 - 2 \beta_{1} - 3 \beta_{2} + 4 \beta_{3} + \beta_{4} ) q^{18}$$ $$+ ( 2 - 3 \beta_{1} - \beta_{4} ) q^{19}$$ $$+ ( 1 + \beta_{3} ) q^{21}$$ $$+ ( -1 + 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{22}$$ $$- q^{23}$$ $$+ ( 1 - \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} ) q^{24}$$ $$+ ( 3 - 4 \beta_{1} - \beta_{3} + \beta_{4} ) q^{26}$$ $$+ ( 4 - 4 \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{4} ) q^{27}$$ $$+ ( \beta_{3} + \beta_{4} ) q^{28}$$ $$+ ( -2 + \beta_{2} + 2 \beta_{3} - 3 \beta_{4} ) q^{29}$$ $$+ ( -2 - \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{31}$$ $$+ ( 2 \beta_{1} + 2 \beta_{3} - \beta_{4} ) q^{32}$$ $$+ ( -3 + \beta_{1} + \beta_{2} - 5 \beta_{3} - 2 \beta_{4} ) q^{33}$$ $$+ ( -1 + 5 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{34}$$ $$+ ( 5 - 4 \beta_{1} - 5 \beta_{2} + 5 \beta_{3} + 3 \beta_{4} ) q^{36}$$ $$+ ( 2 \beta_{1} + 4 \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{37}$$ $$+ ( 4 + 2 \beta_{1} + 3 \beta_{3} + 2 \beta_{4} ) q^{38}$$ $$+ ( 1 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} ) q^{39}$$ $$+ ( -1 + 4 \beta_{1} + 2 \beta_{4} ) q^{41}$$ $$+ ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{42}$$ $$+ ( 1 + 4 \beta_{1} + 3 \beta_{2} + \beta_{3} - \beta_{4} ) q^{43}$$ $$+ ( \beta_{1} + 2 \beta_{2} - 7 \beta_{3} - 3 \beta_{4} ) q^{44}$$ $$+ ( -1 + \beta_{1} ) q^{46}$$ $$+ ( 6 - \beta_{1} - \beta_{2} - 2 \beta_{4} ) q^{47}$$ $$+ ( 2 - 4 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{48}$$ $$+ q^{49}$$ $$+ ( -3 - \beta_{1} + 3 \beta_{2} - 3 \beta_{3} - 4 \beta_{4} ) q^{51}$$ $$+ ( 7 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{52}$$ $$+ ( 3 + \beta_{1} + \beta_{2} + 2 \beta_{4} ) q^{53}$$ $$+ ( 7 - 2 \beta_{1} - 4 \beta_{2} + 8 \beta_{3} + 5 \beta_{4} ) q^{54}$$ $$+ ( -\beta_{2} + \beta_{3} + \beta_{4} ) q^{56}$$ $$+ ( 6 - 6 \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{57}$$ $$+ ( -2 + 3 \beta_{1} - \beta_{2} + \beta_{3} - 3 \beta_{4} ) q^{58}$$ $$+ ( \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{59}$$ $$+ ( 1 + \beta_{1} + 3 \beta_{2} ) q^{61}$$ $$+ ( 4 - \beta_{1} - \beta_{2} + 2 \beta_{3} + 3 \beta_{4} ) q^{62}$$ $$+ ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{63}$$ $$+ ( -3 + \beta_{1} + 2 \beta_{2} - \beta_{4} ) q^{64}$$ $$+ ( -10 + 9 \beta_{1} + 6 \beta_{2} - 7 \beta_{3} - 3 \beta_{4} ) q^{66}$$ $$+ ( 6 + \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{67}$$ $$+ ( -8 + \beta_{1} + 4 \beta_{2} - 3 \beta_{3} - \beta_{4} ) q^{68}$$ $$+ ( -1 - \beta_{3} ) q^{69}$$ $$+ ( -10 + 5 \beta_{1} + \beta_{2} + 4 \beta_{3} + 2 \beta_{4} ) q^{71}$$ $$+ ( 6 - 5 \beta_{1} - 4 \beta_{2} + 6 \beta_{3} + 5 \beta_{4} ) q^{72}$$ $$+ ( 1 + 3 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{73}$$ $$+ ( 1 - \beta_{1} + 3 \beta_{2} - 5 \beta_{3} - 4 \beta_{4} ) q^{74}$$ $$+ ( 3 - 5 \beta_{1} - 3 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{76}$$ $$+ ( -2 + 2 \beta_{2} - \beta_{3} ) q^{77}$$ $$+ ( 3 - 7 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} ) q^{78}$$ $$+ ( 1 + 5 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{79}$$ $$+ ( 4 - 6 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + 5 \beta_{4} ) q^{81}$$ $$+ ( -3 - 5 \beta_{1} - 4 \beta_{3} - 2 \beta_{4} ) q^{82}$$ $$+ ( 2 - 3 \beta_{1} + 4 \beta_{2} - \beta_{3} - \beta_{4} ) q^{83}$$ $$+ ( 2 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{84}$$ $$+ ( -5 \beta_{1} + 2 \beta_{2} - 6 \beta_{3} - 5 \beta_{4} ) q^{86}$$ $$+ ( 8 - 2 \beta_{1} + \beta_{2} - 4 \beta_{3} - 7 \beta_{4} ) q^{87}$$ $$+ ( -7 + 3 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} ) q^{88}$$ $$+ ( -2 \beta_{2} + 5 \beta_{3} - 2 \beta_{4} ) q^{89}$$ $$+ ( \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{91}$$ $$+ ( -\beta_{3} - \beta_{4} ) q^{92}$$ $$+ ( 4 - 4 \beta_{1} - 5 \beta_{2} + \beta_{3} + 3 \beta_{4} ) q^{93}$$ $$+ ( 4 - 3 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{94}$$ $$+ ( 5 + 2 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{96}$$ $$+ ( 1 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + 5 \beta_{4} ) q^{97}$$ $$+ ( 1 - \beta_{1} ) q^{98}$$ $$+ ( -10 + 7 \beta_{1} + 2 \beta_{2} - 8 \beta_{3} - 5 \beta_{4} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5q$$ $$\mathstrut +\mathstrut 3q^{2}$$ $$\mathstrut +\mathstrut 6q^{3}$$ $$\mathstrut +\mathstrut 3q^{4}$$ $$\mathstrut +\mathstrut 7q^{6}$$ $$\mathstrut +\mathstrut 5q^{7}$$ $$\mathstrut +\mathstrut 3q^{8}$$ $$\mathstrut +\mathstrut 5q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$5q$$ $$\mathstrut +\mathstrut 3q^{2}$$ $$\mathstrut +\mathstrut 6q^{3}$$ $$\mathstrut +\mathstrut 3q^{4}$$ $$\mathstrut +\mathstrut 7q^{6}$$ $$\mathstrut +\mathstrut 5q^{7}$$ $$\mathstrut +\mathstrut 3q^{8}$$ $$\mathstrut +\mathstrut 5q^{9}$$ $$\mathstrut -\mathstrut 11q^{11}$$ $$\mathstrut +\mathstrut 14q^{12}$$ $$\mathstrut +\mathstrut 7q^{13}$$ $$\mathstrut +\mathstrut 3q^{14}$$ $$\mathstrut -\mathstrut q^{16}$$ $$\mathstrut -\mathstrut q^{17}$$ $$\mathstrut +\mathstrut 17q^{18}$$ $$\mathstrut +\mathstrut 2q^{19}$$ $$\mathstrut +\mathstrut 6q^{21}$$ $$\mathstrut -\mathstrut 2q^{22}$$ $$\mathstrut -\mathstrut 5q^{23}$$ $$\mathstrut +\mathstrut 12q^{24}$$ $$\mathstrut +\mathstrut 8q^{26}$$ $$\mathstrut +\mathstrut 15q^{27}$$ $$\mathstrut +\mathstrut 3q^{28}$$ $$\mathstrut -\mathstrut 14q^{29}$$ $$\mathstrut -\mathstrut 6q^{31}$$ $$\mathstrut +\mathstrut 4q^{32}$$ $$\mathstrut -\mathstrut 22q^{33}$$ $$\mathstrut +\mathstrut 4q^{34}$$ $$\mathstrut +\mathstrut 28q^{36}$$ $$\mathstrut +\mathstrut q^{37}$$ $$\mathstrut +\mathstrut 31q^{38}$$ $$\mathstrut +\mathstrut 15q^{39}$$ $$\mathstrut +\mathstrut 7q^{41}$$ $$\mathstrut +\mathstrut 7q^{42}$$ $$\mathstrut +\mathstrut 12q^{43}$$ $$\mathstrut -\mathstrut 11q^{44}$$ $$\mathstrut -\mathstrut 3q^{46}$$ $$\mathstrut +\mathstrut 24q^{47}$$ $$\mathstrut +\mathstrut 4q^{48}$$ $$\mathstrut +\mathstrut 5q^{49}$$ $$\mathstrut -\mathstrut 28q^{51}$$ $$\mathstrut +\mathstrut 36q^{52}$$ $$\mathstrut +\mathstrut 21q^{53}$$ $$\mathstrut +\mathstrut 49q^{54}$$ $$\mathstrut +\mathstrut 3q^{56}$$ $$\mathstrut +\mathstrut 15q^{57}$$ $$\mathstrut -\mathstrut 9q^{58}$$ $$\mathstrut -\mathstrut q^{59}$$ $$\mathstrut +\mathstrut 7q^{61}$$ $$\mathstrut +\mathstrut 26q^{62}$$ $$\mathstrut +\mathstrut 5q^{63}$$ $$\mathstrut -\mathstrut 15q^{64}$$ $$\mathstrut -\mathstrut 45q^{66}$$ $$\mathstrut +\mathstrut 35q^{67}$$ $$\mathstrut -\mathstrut 43q^{68}$$ $$\mathstrut -\mathstrut 6q^{69}$$ $$\mathstrut -\mathstrut 32q^{71}$$ $$\mathstrut +\mathstrut 36q^{72}$$ $$\mathstrut +\mathstrut 7q^{73}$$ $$\mathstrut -\mathstrut 10q^{74}$$ $$\mathstrut +\mathstrut 10q^{76}$$ $$\mathstrut -\mathstrut 11q^{77}$$ $$\mathstrut +\mathstrut 8q^{78}$$ $$\mathstrut +\mathstrut 18q^{79}$$ $$\mathstrut +\mathstrut 21q^{81}$$ $$\mathstrut -\mathstrut 33q^{82}$$ $$\mathstrut +\mathstrut q^{83}$$ $$\mathstrut +\mathstrut 14q^{84}$$ $$\mathstrut -\mathstrut 26q^{86}$$ $$\mathstrut +\mathstrut 18q^{87}$$ $$\mathstrut -\mathstrut 41q^{88}$$ $$\mathstrut +\mathstrut q^{89}$$ $$\mathstrut +\mathstrut 7q^{91}$$ $$\mathstrut -\mathstrut 3q^{92}$$ $$\mathstrut +\mathstrut 19q^{93}$$ $$\mathstrut +\mathstrut 14q^{94}$$ $$\mathstrut +\mathstrut 26q^{96}$$ $$\mathstrut +\mathstrut 9q^{97}$$ $$\mathstrut +\mathstrut 3q^{98}$$ $$\mathstrut -\mathstrut 54q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5}\mathstrut -\mathstrut$$ $$2$$ $$x^{4}\mathstrut -\mathstrut$$ $$4$$ $$x^{3}\mathstrut +\mathstrut$$ $$4$$ $$x^{2}\mathstrut +\mathstrut$$ $$3$$ $$x\mathstrut -\mathstrut$$ $$1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} - 2 \nu^{2} - 3 \nu + 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{4} - 2 \nu^{3} - 3 \nu^{2} + 2 \nu + 1$$ $$\beta_{4}$$ $$=$$ $$-\nu^{4} + 2 \nu^{3} + 4 \nu^{2} - 4 \nu - 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$1$$ $$\nu^{3}$$ $$=$$ $$2$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$7$$ $$\beta_{1}$$ $$\nu^{4}$$ $$=$$ $$7$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$8$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$18$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$2$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.79802 1.17316 0.266708 −0.787589 −1.45030
−1.79802 1.59027 1.23287 0 −2.85933 1.00000 1.37931 −0.471046 0
1.2 −0.173158 −1.11762 −1.97002 0 0.193524 1.00000 0.687440 −1.75093 0
1.3 0.733292 2.28713 −1.46228 0 1.67714 1.00000 −2.53886 2.23098 0
1.4 1.78759 −0.0742225 1.19548 0 −0.132679 1.00000 −1.43816 −2.99449 0
1.5 2.45030 3.31444 4.00395 0 8.12135 1.00000 4.91027 7.98549 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$1$$
$$7$$ $$-1$$
$$23$$ $$1$$

## Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4025))$$:

 $$T_{2}^{5}$$ $$\mathstrut -\mathstrut 3 T_{2}^{4}$$ $$\mathstrut -\mathstrut 2 T_{2}^{3}$$ $$\mathstrut +\mathstrut 10 T_{2}^{2}$$ $$\mathstrut -\mathstrut 4 T_{2}$$ $$\mathstrut -\mathstrut 1$$ $$T_{3}^{5}$$ $$\mathstrut -\mathstrut 6 T_{3}^{4}$$ $$\mathstrut +\mathstrut 8 T_{3}^{3}$$ $$\mathstrut +\mathstrut 7 T_{3}^{2}$$ $$\mathstrut -\mathstrut 13 T_{3}$$ $$\mathstrut -\mathstrut 1$$ $$T_{11}^{5}$$ $$\mathstrut +\mathstrut 11 T_{11}^{4}$$ $$\mathstrut +\mathstrut 14 T_{11}^{3}$$ $$\mathstrut -\mathstrut 185 T_{11}^{2}$$ $$\mathstrut -\mathstrut 612 T_{11}$$ $$\mathstrut -\mathstrut 452$$